Puzzle #88: Snowballs

Anticipating a neighbourhood snowball fight, George has several caches of snowballs prepared. The total number of snowballs is even. No cache’s size is evenly divisible by another cache’s size. (For example, if there is a cache that has two snowballs, no other cache can have an even number of snowballs.) What is the minimum number of snowballs that George can have if he has four caches?

Submit your answer to Gene Wirchenko <genew@telus.net>. Your answer should be in the form of a proof. That means to show how your answer must be correct. The deadline is Wednesday, February 11, 2015> at noon Pacific Time. I will post the answer shortly after.